2. Function Composition Given f(x) = 2x + 2 and g(x) = 2, find f ºg(x). f ºg(x)=f(g(x)Start on the inside. f(g(x)) g(x) = 2, so replace it. f(g(x)) = f(2) = 2(2) + 2 = 6 f ºg(x) = 6

3. Another one… Given g(x) = 2x - 5 and f(x) = x + 1, find f ºg(x). f ºg(x) = f(g(x)) g(x) = 2x - 5 so replace it. f(g(x)) = f(2x - 5) Now replace x with 2x - 5 in f(x). f(2x - 5) = (2x - 5) + 1 = 2x - 5 + 1 = 2x – 4 = f ºg(x)

4. Function Composition Given g(x) = 2x - 5 and f(x) = x + 1, find f ºg(x). Find g º f(x).Well g º f(x)=g(f(x)). f(x) = x + 1 so replace it. g(x + 1). g(x + 1) = 2(x + 1) - 5 = 2x+2 - 5 So g º f(x)=2x - 3

5. Function Composition f ºg(x)= 2x – 4 g º f(x)=2x – 3 What do you notice? Function Composition is not commutative, that is

6. Function Composition Given f(x) = x 2 + x and g(x) = x - 4, find f ºg(x) and g ºf(x). f(g(x)) = f(x - 4) = (x - 4) 2 + (x - 4) = x 2 - 8x + 16 + x - 4 = x 2 - 7x+12 f ºg(x) = x 2 - 7x + 12

7. Function Composition Given f(x) = x 2 + x and g(x) = x - 4, find f ºg(x) and g ºf(x). g(f(x)) = g(x 2 + x) = x 2 + x - 4

8. Function Composition: Values Given f(x) = 2x + 5 and g(x) = 8 + x, find f º g(-5). f º g(-5) = f(g(-5)) Start with innermost parentheses: g(-5) = 8 + -5 = 3. So replace g(-5) with 3 and we get f(3) = 2(3) + 5 = 6 + 5 = 11 f ºg(-5) = 11

9. Function Composition Given f(x) = 2x + 5 and g(x) = 8 + x, find g º f(-5). g º f(-5) = g(f(-5) Start in the innermost parentheses: f(-5) = 2(-5) + 5 = -10 + 5 = 5 Replace f(-5) with 5 and we have g(5) = 8 + 5 = 13. g ºf(-5) = 13

10. Function Composition Assignment: Worksheet

11. Function Inverse Quick review. {(2, 3), (5, 0), (-2, 4), (3, 3)} Domain & Range = ? Inverse = ? D = {2, 5, -2, 3} R = {3, 0, 4}

12. Function Inverse {(2, 3), (5, 0), (-2, 4), (3, 3)} Inverse = switch the x and y, (domain and range) I = {(3, 2), (0, 5), (4, -2), (3, 3)}

13. Function Inverse {(4, 7), (1, 4), (9, 11), (-2, -1)} Inverse = ? I = {(7, 4), (4, 1), (11, 9), (-1, -2)}

14. Function Inverse Now that we can find the inverse of a relation, let’s talk about finding the inverse of a function. What is a function? a relation in which no member of the domain is repeated.

15. Function Inverse To find the inverse of a function we will still switch the domain and range, but there is a little twist … We will be working with the equation.

16. Function Inverse So what letter represents the domain? x So what letter represents the range? y

17. Function Inverse So we will switch the x and y in the equation and then resolve it for … y.

18. Function Inverse Find the inverse of the function f(x) = x + 5. Substitute y for f(x). y = x + 5. Switch x and y. x = y + 5 Solve for y. x - 5 = y

19. Function Inverse So the inverse of f(x) = x + 5 is y = x - 5 or f(x) = x - 5.

20. Function Inverse Given f(x) = 3x - 4, find its inverse (f -1 (x)). y = 3x - 4 switch. x = 3y - 4 solve for y. x + 4 = 3y y = (x + 4)/3

21. Function Inverse Given h(x) = -3x + 9, find it’s inverse. y = -3x + 9 x = -3y + 9 x - 9 = -3y (x - 9) / -3 = y

22. Function Inverse Given Find the inverse.

23. Function Inverse

24. Function Inverse 3x = 2y + 5 3x - 5 = 2y

25. Function Inverse Given f(x) = x 2 - 4 y = x 2 - 4 x = y 2 - 4 x + 4 = y 2

26. Function Inverse x + 4 = y 2

27. Function Inverse Assignment pg. 532 15 - 33 odd